I am pretty sure it's metastatic lung cancer. The clues are various mention of lungs and the stage 4 in the beginning of the article. [I was wrong it was breast cancer]
If you think the piece is to divulge which condition her mother had (which, to be frank, was pretty obvious from reading the context clues), you missed the point.
The opening is "I looked in her emails to find whether she lied", but it then goes into the entire life story of the author and their mother type of article, which... I just don't have the time for or interest in. More of a vacation "Reader's Digest" type story.
The condition is never named in the article.